Q: Imaginary Part of meromorphic functions I'm currently studying DeBranges book about Hilbert Spaces of Entire functions. This questions deals with problem 60 on page 61 (page 71 in the pdf on his website DeBranges Website).
Let $s_n,t_n$ be (potentially infinite unbounded) positive sequences such that $s_n<t_n<s_{n+1}<t_{n+1}$ (interlacing property) and let
$$f(z)=\prod_{n=1}^\infty\frac{1-z/t_n}{1-z/s_n},\qquad z\neq s_n.$$
Then $f$ defines a meromorphic function with only simple, real poles and zeros on $\mathbb{C}$ which is real for real $z$. I want to show that it is in fact real if, and only if, $z$ is real. I tried working with a few examples and found the following problems:
Let's go finite first. If we disregard the poles, then there is an easy counterexample:
$$f(z)=(z-1)(z+1)=z^2-1,\qquad \text{then }f(\mathrm{i})=-2\in\mathbb{R}.$$
Okay, the poles are relevant here. In particular
$$f(z)=\frac{z^2-1}{z},\qquad \text{then }\Im(f(a+\mathrm{i}b))=\frac{b(a^2+1+b^2)}{a^2+b^2}\neq 0\quad \text{for }b\neq 0.$$
But why? What part of the pole property makes this work, if we multiply by a pole? Is it the interlacing part? Again, but why?
I tried computing the Imaginary part of finite products - there it actually works out. By partial fraction decomposition we obtain the following sum
$$f_n(z)=h+\sum_{k=1}^n\frac{G_n(s_k)}{F_n'(s_k)(z-s_k)},\qquad\text{where } h\in\mathbb{R},\;G_n(z)=\underbrace{\prod_{k=1}^n (1-z/t_k)}_{\text{numerator}},\;F_n(z)=\underbrace{\prod_{k=1}^n(1-z/s_k)}_{\text{denominator}}$$
by computing the residues. The interlacing property of the zeros and poles guarantees that $G_n/F_n'$ doesn't change sign and thus we get a positive imaginary part for $z\in\mathbb{C}^+$. The problem arises now if we take the limit to infinity, because then the strict inequality might become an equality. So once again I don't see a reason why the imaginary part shouldn't be zero.
I would much appreciate any input. My only idea would be to do a partial fraction decomposition on all of $f$, but there I don't see why it should converge at all. Is there some result I don't know which allows me to do the decomposition on all potentially infinite poles?
